Texans will go to the polls on November 5 to vote on nine proposed amendments to the state constitution, and some areas are also holding local government elections. It is the first statewide vote since it became mandatory in Texas to show a government-issued photo ID at polling places.

Some critics of the new law believe that women who have changed their name, for example after marriage or divorce, may be discouraged from voting or run into difficulties while trying.

If a prospective voter's name does not exactly match a name on the list of registered voters, it is up to the election officer at the polling station to determine whether the name is "substantially similar". If so, the person will be allowed to cast a ballot after signing an affidavit attesting to his or her identity. Those without approved photo ID can vote "provisionally" and then have six days after election day to present acceptable proof to a county registrar.

Early voting is already under way. Local media reported that a judge in Corpus Christi and Wendy Davis, a Democratic state senator who is running for governor, had to sign affidavits at the polling station because the names on their driving licenses and voter registration cards were not identical.

"There’s a tremendous concern it will create a problem for women who have been legally voting for years to be able to vote … and that they may be surprised by it," Davis told the Fort Worth Star-Telegram. "I hope the word will get out. I hope we will continue to see women vote as they have in Texas."

Tanene Allison, communications director of the Texas Democratic party, told the Guardian that the measure is "definitely having an impact on women voters" but the full implications are not yet clear, as this is the initial stage of an election that is expected to have a low turnout.

Democrats, who view a high female turnout next year as crucial to Davis's prospects, have accused Republicans of conducting a "war on women", not least thanks to the introduction of strict new abortion rules that were partially blocked on Monday by a federal judge.

Alicia Phillips Pierce, communications director for the Texas secretary of state, John Steen, told the Guardian: "We've had a week of early voting and we haven't seen anyone turned away from the polls for a similar name. On the whole, we're not hearing any reports of problems. More than 100,000 Texans have voted and things seem to be going smoothly."

Linda Krefting, president of the non-partisan League of Women Voters of Texas, said her organisation had opposed the introduction of the law but is now focused on encouraging people to vote by helping them understand the requirements.

She told the Guardian that it remains to be seen whether voters would be treated fairly in all precincts across the state. Krefting also raised the possibility that electoral workers could be more familiar with Anglo nicknames than variants that may be used by minorities.

Voters must present one of seven forms of ID, such as a driver's license, a passport or a concealed handgun permit. Alternatively, the state has introduced an Election Identification Certificate, which is free and available to eligible residents from driver's license offices. But only 88 had been issued statewide by Monday night.

Postal voters and disabled people can circumvent the ID requirement.

The Republican-dominated Texas legislature passed the law in 2011, with Davis offering the amendment that allowed for affidavits. But the law was blocked by a federal court last year, which ruled that the costs and travel burden of obtaining ID were potentially unfair on the poor and could reduce turnout among minorities.

However, on June 25 this year the supreme court struck down the element of the Voting Rights Act that required states with a history of racially-discriminatory voting practices to gain federal approval before making changes.

On the same day, Greg Abbott, the Texas attorney general, announced that Texas's ID law – and redrawn electoral boundaries also halted by a federal court for being discriminatory – were effective immediately.

The Department of Justice and civil rights groups have issued legal challenges against both. The redistricting case will be heard in July, 2014. No schedule has yet been set for the voter ID case.

Proponents of voter ID laws argue that they help prevent fraud. Opponents claim that instances of such abuse are negligible and that the true motivation is to disenfranchise people on low incomes and minorities, who often do not have ID or the means to get it easily. Those groups tend to vote Democrat.

The Department of Justice contends that Texas' new laws were adopted with the purpose of "denying or abridging the right to vote on account of race, colour or membership in a language minority group" and said in a legal filing that in comparison with white voters the ID rule will "disproportionately affect Hispanic and African-American Texans, who are disproportionately poor and disproportionately lack access to transportation."

It cited data produced by Texas which suggests that about 6% of the state's registered voters – roughly 800,000 people – lacked suitable ID.

Abbott, a Republican, is set to battle Davis in the Texas gubernatorial election in November, 2014. He told the Dallas Morning News last month that he wants to preserve "the integrity of the election system" and that "the horrors that are raised about voter suppression have proven to be untrue in real life, on the ground." Abbott's office has prosecuted 66 cases related to voter irregularities since 2004, half involving postal ballot violations. Only three involved voter impersonation.

Republican-controlled states have increasingly attempted to introduce voter ID laws in recent years. The Department of Justice is also suing North Carolina over its new law, which requires photo identification, restricts early voting and ends same-day registration. A North Carolina conservative activist resigned last week after telling The Daily Show that the rules would "kick the Democrats in the butt" and making reference to "lazy blacks who want the government to give them everything."